?Design contrasts to compare chewy vs crispy, and expensive with inexpensive

33 (and also compare a contrast of )
Consider a taste test experiment of six types of cookies
1 (brand A,chewy,expensive)
2 (brand A,crispy,expensive)
3 (brand B,chewy,inexpensive)
4 (brand B,crispy,inexpensive)
5 (brand C,chewy,expensive)
6 (brand D,crispy,inexpensive)

At random, 20 different rater are assigned to each cookie.

Design contrasts to compare chewy vs crispy, and expensive with inexpensive.

Are the contrasts orthogonal? Show.

For the example in class with 3 levels: control, trt1 and trt2, we used contrasts w1=(1,-0.5,-0.5) and w2=(0,1,-1). Now assume the sample sizes were n1=10, n2=7, n3=8. Rescale w1 and/or w2 so that they are still orthogonal and still make the same relative comparisons, i.e. compare control to the two trts and compare trt1 to trt2.

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